The distributions markets actually produce

Most of finance’s closed-form results assume returns are normally distributed. They are a useful first approximation and wrong in exactly the places that matter most: the tails.

The normal distribution

A normal variable with mean $\mu$ and standard deviation $\sigma$ has density

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

Rules of thumb worth memorising: about 68% of outcomes fall within $1\sigma$, 95% within $1.96\sigma$, and 99.7% within $3\sigma$. Under normality a $5\sigma$ down day should happen roughly once every 14,000 years of trading.

The normal earns its place through the central limit theorem: the sum of many independent, finite-variance shocks is approximately normal regardless of what the individual shocks look like. Monthly returns, being sums of daily ones, look more normal than daily returns.

The lognormal distribution

Prices cannot go below zero, so they cannot be normal. If the log return $\ln(S_T/S_0)$ is normal, the price $S_T$ is lognormal. This is the assumption behind geometric Brownian motion and Black–Scholes.

A lognormal is right-skewed. If $\ln S_T \sim \mathcal{N}(m, s^2)$ then

$$\mathbb{E}[S_T] = e^{m + s^2/2}, \qquad \text{median}(S_T) = e^{m}$$

The mean sits above the median: a few large winners pull the average up while the typical outcome is lower. This gap is the root of the volatility drag you will meet again in the Kelly lesson.

Fat tails

Real returns have far more extreme observations than a normal predicts. Two summary statistics catch this:

• Skewness $= \mathbb{E}[(X-\mu)^3]/\sigma^3$ measures asymmetry. Equity indices tend to be negatively skewed: slow grinds up, sudden falls.
• Kurtosis $= \mathbb{E}[(X-\mu)^4]/\sigma^4$ measures tail weight. A normal has kurtosis 3; daily equity and crypto returns routinely show 6, 10, or much more. “Excess kurtosis” subtracts the 3.

A common fat-tailed model is Student’s t with $\nu$ degrees of freedom. As $\nu \to \infty$ it becomes the normal; at $\nu$ around 3 to 5 it fits daily returns reasonably well. For $\nu \le 4$ its kurtosis is infinite, which is a polite way of saying “sample kurtosis will never settle down”.

Checking a distribution

1. Plot a histogram of returns against a normal with the same mean and variance. Look at the tails, not the middle.
2. Use a QQ plot: sample quantiles against normal quantiles. Points that bend away from the diagonal at both ends signal fat tails.
3. Compute rolling volatility. If it moves a lot, the unconditional distribution will be fat-tailed even if each regime is not.